Explore the vast range of titles available.

A C∞ smooth surface diffeomorphism admits an SRB measure if and only if the set {x,lim supn1nlog∥dxfn∥>0} has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost everywhere. We also obtain similar results for Cr surface diffeomorphisms with +∞>r>1.

We unveil instances where nonautonomous linear systems manifest distinct nonuniform μ -dichotomy spectra despite admitting nonuniform ( μ , ε ) -kinematic similarity. Exploring the theoretical foundations of this lack of invariance, we discern the pivotal influence of the parameters involved in the property of nonuniform μ -dichotomy such as in the notion of nonuniform ( μ , ε ) -kinematic similarity. To effectively comprehend these dynamics, we introduce the stable and unstable optimal ratio maps, along with the ε -neighborhood of the nonuniform μ -dichotomy spectrum. These new concepts provide a framework for understanding scenarios governed by the noninvariance of the nonuniform μ -dichotomy spectrum.

Consider an Itô process X satisfying the stochastic differential equation dX = a(X) dt + b(X) dW where a, b are smooth and W is a multidimensional Brownian motion. Suppose that Wn has smooth sample paths and that Wn converges weakly to W. A central question in stochastic analysis is to understand the limiting behavior of solutions Xn to the ordinary differential equation dXn = a(Xn) dt + b(Xn) dWn. The classical Wong-Zakai theorem gives sufficient conditions under which Xn converges weakly to X provided that the stochastic integral ∫ b(X) dW is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of ∫ b(X) dW depends sensitively on how the smooth approximation Wn is chosen. In applications, a natural class of smooth approximations arise by setting Wn (t) = n-1/2 $\\smallint _0^{nt}\\upsilon o{\\phi _s}ds$ where ɸt is a flow (generated, e.g., by an ordinary differential equation) and υ is a mean zero observable. Under mild conditions on ɸt, we give a definitive answer to the interpretation question for the stochastic integral ∫ b(X) dW. Our theory applies to Anosov or Axiom A flows ɸt, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on ɸt. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.

In this paper, we mainly focus on the set of non-dense points of a dynamical system. We study the distributional chaos in such set. As for a mixing expanding map or a transitive Anosov diffeomorphism on a compact connected manifold, we prove that DC1 chaos can occur in a zero topological entropy subset of the intersection of the set of recurrent points and the set of the non-dense points. Also, for such dynamical systems, strongly distributional chaos (which is stronger than DC1 chaos) can occur in a zero topological entropy subset of the set of non-recurrent points. Besides, when we divide the total space into six layers according to the different statistical structures, similar results can appear in every layer. Our results can also be applied to mixing subshifts of finite type, β -shifts, homoclinic classes and C 1 + α diffeomorphisms preserving a weakly mixing hyperbolic ergodic measure.

We prove that smoothness of nonautonomous linearization is of class C 2 . Our approach admits the existence of stable and unstable manifolds determined by a family of nonautonomous hyperbolicities, including the non uniform exponential case, while for the classic exponential dichotomy we obtain the same class of differentiability except for a zero Lebesgue measure set. Moreover, our goal is reached without spectral conditions.

For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.

Given a continuous linear cocycle$\\mathcal {A}$over a homeomorphism f of a compact metric space X , we investigate its set$\\mathcal {R}$of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set$\\mathcal {R}$is of first Baire category (i.e., meager) in X , unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.

We prove that generic fiber-bunched and Hölder continuous linear cocycles over a non-uniformly hyperbolic system endowed with a $u$-Gibbs measure have simple Lyapunov spectrum. This gives an affirmative answer to a conjecture proposed by Viana in the context of fiber-bunched linear cocycles.

We prove that a partially hyperbolic attractor for a C 1 vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is C 2 , and the center bundle admits the sectionally expanding condition w.r.t. Gibbs u -states, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms.

This article studies the Bowen-Margulis-Sullivan (BMS) measures on non-compact manifolds without conjugate points. The finiteness of this measure on the unit tangent space indicates some important dynamical properties. Under the assumptions of uniform visibility axiom and Axiom 2, we give a criterion when the BMS measure is finite.